On the interplay between Noether's theorem and the theory of adiabatic invariants
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866916152991547392 |
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| author | Gourieux, Thierry Leone, Raphaël |
| author_facet | Gourieux, Thierry Leone, Raphaël |
| contents | This article focuses on an important quantity that will be called the Rund-Trautman function. It already plays a central role in Noether's theorem since its vanishing characterizes a symmetry and leads to a conservation law. The main aim of the paper is to show how, in the realm of classical mechanics, an 'almost' vanishing Rund-Trautman function accompanying an 'almost' symmetry leads to an 'almost' constant of motion within the adiabatic assumption, that is, to an adiabatic invariant. To this end, the Rund-Trautman function is first introduced and analysed in detail, then it is implemented for the general one-dimensional problem. Finally, its relevance in the adiabatic context is examined through the example of the harmonic oscillator with a slowly varying frequency. Notably, for some frequency profiles, explicit expansions of adiabatic invariants are derived through it and an illustrative numerical test is realized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_08853 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the interplay between Noether's theorem and the theory of adiabatic invariants Gourieux, Thierry Leone, Raphaël Classical Physics This article focuses on an important quantity that will be called the Rund-Trautman function. It already plays a central role in Noether's theorem since its vanishing characterizes a symmetry and leads to a conservation law. The main aim of the paper is to show how, in the realm of classical mechanics, an 'almost' vanishing Rund-Trautman function accompanying an 'almost' symmetry leads to an 'almost' constant of motion within the adiabatic assumption, that is, to an adiabatic invariant. To this end, the Rund-Trautman function is first introduced and analysed in detail, then it is implemented for the general one-dimensional problem. Finally, its relevance in the adiabatic context is examined through the example of the harmonic oscillator with a slowly varying frequency. Notably, for some frequency profiles, explicit expansions of adiabatic invariants are derived through it and an illustrative numerical test is realized. |
| title | On the interplay between Noether's theorem and the theory of adiabatic invariants |
| topic | Classical Physics |
| url | https://arxiv.org/abs/2012.08853 |